Harmonic Mean Calculator
Calculate the harmonic mean of positive numbers with step-by-step explanations and mean comparisons
Enter Your Numbers
Please enter at least 2 valid positive numbers
Example Calculation
Speed Average Example
Numbers: 3, 4, 6, 12
Reciprocals: 1/3, 1/4, 1/6, 1/12
Sum: 0.333 + 0.250 + 0.167 + 0.083 = 0.833
Harmonic Mean: 4 ÷ 0.833 = 4.8
Two Numbers Formula
For x = 2, y = 8:
H = 2 × x × y / (x + y)
H = 2 × 2 × 8 / (2 + 8)
H = 32 / 10 = 3.2
Formulas
General Formula
H = n / Σ(1/xᵢ)
Two Numbers
H = 2xy / (x + y)
Three Numbers
H = 3xyz / (xy + yz + zx)
Applications
Finance: P/E ratio calculations for stock indices
Physics: Average speed for round trips
Electronics: Parallel resistance calculations
Geometry: Triangle incircle radius calculations
Understanding the Harmonic Mean
What is the Harmonic Mean?
The harmonic mean is one of the three Pythagorean means (along with arithmetic and geometric means). It's defined as the reciprocal of the arithmetic mean of the reciprocals of the given numbers.
Key Properties
- •Only defined for positive numbers
- •Always less than or equal to arithmetic and geometric means
- •Heavily influenced by small values
- •Appropriate for rates and ratios
Calculation Steps
- 1.Count the total number of values (n)
- 2.Calculate the reciprocal of each number (1/x)
- 3.Add all reciprocals together
- 4.Divide n by the sum of reciprocals
When to Use Harmonic Mean
- • Average rates (speed, flow rates)
- • Financial ratios (P/E ratios)
- • Parallel circuits (resistance, capacitance)
- • Any situation involving reciprocal relationships
Real-World Example: Average Speed
If you drive 60 km at 30 km/h and then 60 km at 60 km/h, what's your average speed?
Incorrect (Arithmetic Mean)
(30 + 60) ÷ 2 = 45 km/h ❌
This ignores the time spent at each speed
Correct (Harmonic Mean)
2 × 30 × 60 ÷ (30 + 60) = 40 km/h ✅
This accounts for equal distances at different speeds